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Properties

Label 1-4235-4235.1042-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.702 - 0.712i$
Analytic cond. $455.113$
Root an. cond. $455.113$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.633 − 0.774i)2-s + (−0.587 − 0.809i)3-s + (−0.198 + 0.980i)4-s + (−0.254 + 0.967i)6-s + (0.884 − 0.466i)8-s + (−0.309 + 0.951i)9-s + (0.909 − 0.415i)12-s + (−0.113 − 0.993i)13-s + (−0.921 − 0.389i)16-s + (0.825 − 0.564i)17-s + (0.931 − 0.362i)18-s + (−0.941 + 0.336i)19-s + (0.755 + 0.654i)23-s + (−0.897 − 0.441i)24-s + (−0.696 + 0.717i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.633 − 0.774i)2-s + (−0.587 − 0.809i)3-s + (−0.198 + 0.980i)4-s + (−0.254 + 0.967i)6-s + (0.884 − 0.466i)8-s + (−0.309 + 0.951i)9-s + (0.909 − 0.415i)12-s + (−0.113 − 0.993i)13-s + (−0.921 − 0.389i)16-s + (0.825 − 0.564i)17-s + (0.931 − 0.362i)18-s + (−0.941 + 0.336i)19-s + (0.755 + 0.654i)23-s + (−0.897 − 0.441i)24-s + (−0.696 + 0.717i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.702 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.702 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.702 - 0.712i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.702 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3349595496 - 0.8006793605i\)
\(L(\frac12)\) \(\approx\) \(0.3349595496 - 0.8006793605i\)
\(L(1)\) \(\approx\) \(0.5205009816 - 0.3262134024i\)
\(L(1)\) \(\approx\) \(0.5205009816 - 0.3262134024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.633 - 0.774i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.113 - 0.993i)T \)
17 \( 1 + (0.825 - 0.564i)T \)
19 \( 1 + (-0.941 + 0.336i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + (0.610 + 0.791i)T \)
31 \( 1 + (0.736 + 0.676i)T \)
37 \( 1 + (-0.491 - 0.870i)T \)
41 \( 1 + (-0.998 + 0.0570i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.931 + 0.362i)T \)
53 \( 1 + (0.389 + 0.921i)T \)
59 \( 1 + (-0.998 - 0.0570i)T \)
61 \( 1 + (0.774 + 0.633i)T \)
67 \( 1 + (-0.540 + 0.841i)T \)
71 \( 1 + (-0.0285 - 0.999i)T \)
73 \( 1 + (-0.226 - 0.974i)T \)
79 \( 1 + (-0.985 - 0.170i)T \)
83 \( 1 + (-0.856 - 0.516i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.441 - 0.897i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.5555831273736391582794055416, −17.28353216248771370144294820226, −17.12851883415507418245329105862, −16.62781352331962906093047339167, −15.786551498918866040169153212, −15.21708632176424175247651249834, −14.7028468184366117720642370588, −13.98592375638902317333115255558, −13.13179200393325877930619329005, −12.08646432121303796470722568822, −11.47158537895524488634717822965, −10.71335102199580865109167424048, −10.03641716405607663612562321027, −9.64852447096138328274591685333, −8.572990503105302981996270168893, −8.41179266691397468944304715959, −7.114645256605306838758959707173, −6.537182875910818303618261966626, −5.99225838332438397582444406446, −5.06260605874097325406437733813, −4.54278066141311424561595235012, −3.78814454057521024521100820447, −2.573420290616878922708023048854, −1.490261514875369312519717954644, −0.54716284939260730886268994437, 0.317187309745975654527681385, 1.11851928223154844811916002972, 1.754769989242788759898818635296, 2.817503216173393042699786127972, 3.2718103618659073277057211058, 4.51340431395369979678676521869, 5.24313967849387241457394490801, 6.04662936143435418426303338214, 7.121373470005818915954079568511, 7.41746778721957645118879029050, 8.372925983415258355918021756225, 8.792676404885155414507447573163, 9.96500226341153456644988087940, 10.4663289167719003270470002584, 11.032403305943574687925538873490, 11.95910053535720514345703752857, 12.27955640983436195752145569499, 12.97179629630708898243024351682, 13.59315257828329244520190962649, 14.3399851923413192311402410977, 15.42202724043711444192075044457, 16.19177482834185980928746834008, 17.00924398920660016150806856614, 17.28559499452283171286008410982, 18.15274417379374848783682345724

Graph of the $Z$-function along the critical line